To solve the problem using a factorial method, let's break it down step by step:

### Step 1: Total Number of 5-Digit Numbers
We are forming 5-digit numbers using the digits \(0, 1, 2, 3, 4\). 

The first digit (the most significant digit) cannot be \(0\), while the remaining digits can be any of \(0, 1, 2, 3, 4\).

- For the first digit, we have 4 choices: \(1, 2, 3, 4\).
- For the remaining four digits, each digit has 5 choices: \(0, 1, 2, 3, 4\).

The total number of possible 5-digit numbers is:
\[
4 \times 5^4 = 4 \times (5 \times 5 \times 5 \times 5) = 4 \times 625 = 2500
\]

### Step 2: Counting the Favorable Outcomes
We need to find the number of 5-digit numbers where the digit \(2\) is in the tens place and the digit \(0\) is in the units place.

- The tens place is fixed as \(2\).
- The units place is fixed as \(0\).
- The first digit (thousands place) has 4 choices: \(1, 2, 3, 4\).
- The second digit has 5 choices: \(0, 1, 2, 3, 4\).
- The third digit has 5 choices: \(0, 1, 2, 3, 4\).

The number of favorable outcomes is calculated by multiplying these choices together:
\[
4 \times 5 \times 5 = 100
\]

### Step 3: Calculate the Probability
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:
\[
\text{Probability} = \frac{100}{2500} = \frac{1}{25}
\]

### Final Answer
Using the factorial method, the probability of getting \(2\) in the tens place and \(0\) in the units place is \(\boxed{\frac{1}{25}}\).

Would you like to delve deeper into the factorial method or have any questions?

Here are some related questions:
1. How would the probability change if the digits were allowed to repeat?
2. What is the probability of the first digit being \(3\) and the last digit being \(1\)?
3. Can you calculate the probability using permutations instead of combinations?
4. What is the probability of all digits being odd numbers?
5. How many 5-digit numbers can be formed if each digit must be different?

**Tip:** Factorials are very useful for calculating probabilities, especially when dealing with permutations and combinations in probability theory.

Learn how to calculate the probability of specific digit placements in 5-digit numbers using the factorial method. This problem involves combinatorial calculations and probability theory. Suitable for students in Grades 10-12.

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